The control of waves using periodic structures is crucial for
modern optical, electromagnetic and acoustic devices such as diffraction
gratings, filters, photonic crystals and meta-materials, solar cells, and
absorbers. We present a high-order accurate boundary-based numerical
solver
for three-dimensional (3D) frequency-domain scattering from a
doubly-periodic lattice of objects. We focus on the case of axisymmetric
objects, and handle both the acoustic and electromagnetic cases. We
combine
the method of fundamental solutions with a new periodizing scheme, and
with
various fast algorithms such as the fast multiple method, and so-called
"skeletonization". Our scheme has exponential convergence property, avoids
singular quadratures, periodic Green's functions, and lattice sums, and
its
convergence rate is unaffected by resonances within obstacles. We also
discuss new methods for handling corner singularities.